There is a matching lower bound: for , and, using (consequence of Stirling formula), we obtain . Therefore, with the upper bound:

.

However, I suspect that can fluctuate when is between two successive points of that would be "far away from each other". Allowing for arbitrary seems too weak an assumption to me: it can be an incredibly "hollow" sequence...

Can you prove the theorem for ? Or for any other sequence that grows faster than linearly? (I would be curious of what method you would use)

Now I have a proof that fluctuations between points of can break the conjecture. I shall provide a family of examples where .

Note that is equivalent to .

Suppose is such that where and (thus is a gap in , thought of as a large gap). More precise conditions will come later.

We have the following upper bound that consists in taking all terms of the series outside the gap:

.

For the first sum, note that the largest term is the last one (each term is times the previous one), so that the sum is less than .

For the second sum, I factorize by the first term and use a geometric series to bound the second factor:

.

Summarizing, we have .

I shall now give conditions that ensure or where is either of the two terms of the upper bound, and the limit is taken along some sequence (corresponding to some intervals ).

First one: . (Using ), hence as soon as and . (Indeed, ) * I forgot , which also tends to 0 (faster) under this condition.

Second one: if and (using Stirling estimate: and ), hence this term equals , so that it tends to as soon as and .

Since , we deduce that where the bound corresponds to a limit taken along sequences of that are in gaps of such that (" " meaning "negligible compared to"). Such sequences of 's exist when grows very fast (we must have where and are successive terms in the sequence): for instance, so and is such an example.

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However, I would still be interested in knowing how you would deal with if you know that.

Marvellous! You're a champ. I wouldn't have known how to do this myself.

I do not know how to deal with the specific case you give. I only had some empirical reasons to believe the conjecture was true - I did some tests with Mathematica and it seemed to hold with some very sparse sets...